Current modulation by voltage control of the quantum phase in ...

PHYSICAL REVIEW B 86, 045418 (2012)

Current modulation by voltage control of the quantum phase in crossed graphene nanoribbons K. M. Masum Habib* and Roger K. Lake† Department of Electrical Engineering, University of California, Riverside, CA 92521-0204 (Received 3 September 2011; revised manuscript received 24 May 2012; published 11 July 2012) A relative rotation of 90◦ between two graphene nanoribbons (GNRs) creates a crossbar with a nanoscale overlap region. Calculations, based on the first principles density functional theory (DFT) and the nonequilibrium Green’s function (NEGF) formalism, show that the electronic states of the individual GNRs of an unbiased crossbar are decoupled from each other similar to the decoupling that occurs in twisted bilayer graphene. Analytical calculations, based on Fermi’s golden rule, reveal that the decoupling is a consequence of the cancellation of quantum phases of the electronic wave functions of the individual GNRs. As a result, the inter-GNR transmission is strongly suppressed over a large energy window. An external bias applied between the GNRs changes the relative phases of the wave functions resulting in modulation of the transmission and current by several orders of magnitude. A built-in potential between the two GNRs can lead to a large peak-to-valley current ratio (>1000) resulting from the strong electronic decoupling of the two GNRs that occurs when they are driven to the same potential. Current switching by voltage control of the quantum phase in a graphene crossbar structure is a novel switching mechanism. It is robust even with an overlap of ∼1.8 nm × 1.8 nm that is well below the smallest horizontal length scale envisioned in the international technology roadmap for semiconductors (ITRS). DOI: 10.1103/PhysRevB.86.045418

PACS number(s): 72.80.Vp

I. INTRODUCTION

Lack of a band gap in graphene1,2 is one of the challenges for achieving high ON/OFF current ratios in graphene field effect transistors (FETs). The most obvious way to circumvent this problem is to open a band gap, e.g., by using chemical doping,3 creating nanoribbons,4–6 or by applying a vertical electric field in bilayer graphene.7–9 However, it is difficult to create a sufficiently large band gap without degrading the electronic properties of graphene. Another way is to utilize the unique properties of graphene in alternative FET architectures.10–13 A highly nonlinear current-voltage relationship can be obtained in a graphene-insulator-graphene p-n junction.14 Some devices exhibiting negative differential resistance (NDR) have been proposed.15–19 However, most of these devices have relatively complex architectures,12,13,15–17 limited scalability,10 or low on-off or peak-to-valley current ratios.17–19 In this work, we unveil a current switching mechanism in graphene crossbars in which the current can be modulated by several orders of magnitude. This switching mechanism is based on voltage control of the relative phases of the electronic wave functions of two crossed graphene nanoribbons. It does not rely on a band gap, and it is not based on tunneling through or over a potential barrier. It is relatively independent of temperature. It is robust even when the overlap of the active region is scaled down to ∼1.8 nm × 1.8 nm. This length scale is well below any horizontal scale envisioned in the ITRS.20 Interest in twisted, or misoriented, layers of graphene was recently motivated by the need to understand the electronic properties of multilayer graphene furnace grown on the C face of SiC.21 Experimental analysis showed that the layers tended to be rotated with respect to each other at certain angles corresponding to allowed growth orientations with respect to the SiC substrate.22 Calculations, based on density functional theory,21–24 empirical tight-binding,25 and continuum26 models for such rotated bilayers found linear dispersion near the K points. A recent experiment showed that 1098-0121/2012/86(4)/045418(9)

in twisted bilayer graphene for twist angles greater than ∼3◦ , the low-energy carriers behave as massless Dirac fermions with a reduced Fermi velocity compared to that of single-layer graphene, and that for twist angles greater than 20◦ , the layers are effectively decoupled and act as independent layers.27 A vertical electric field in a twisted bilayer graphene can couple the layers28 and reduce the Fermi velocity.29 A recent study of the conductivity between two infinite rotated sheets of graphene found enhanced conductance at commensurate angles with relatively small unit cells and negative differential resistance at small biases.30 Although the physics of the decoupled layers in twisted bilayer graphene has been studied extensively, it is not clear if these properties still hold in the limit of nanoscaled device dimensions, for example, in the twisted bilayer that occurs in the overlap region of two crossed GNRs fabricated by unzipping two carbon nanotubes.31 Botello-Méndez et al. very recently addressed this issue performing both DFT and empirical tight-binding calculations of the transmission across and through crossed graphene nanoribbons.32 Crossed armchair-zigzag (AZ) GNRs and crossed zigzag-zigzag GNRs were considered. Most relevant to our work, was their study of crossed AZ GNRs, approximately 5-nm wide, aligned in AB stacking at right angles and then rotated. The minimum in the interlayer transmission between the armchair GNR (aGNR) and the zigzag GNR occurred when the angle of intersection was 60◦ . This is equivalent to the 90◦ angle of intersection between two aGNRs, which is the system that we consider. In this work, we analyze the physical mechanism of the interlayer coupling at the nanoscale and its dependence on the potential difference between the two layers, and we show that it can be exploited for current switching by voltage control of the wave function phase. The model structure shown in Fig. 1 consists of two armchair GNRs with one placed on top of the other at right angles forming a GNR crossbar (xGNR). In this case, the overlap region of the xGNR, which is neither AA nor AB stacking, is a twisted bilayer with an area of

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©2012 American Physical Society

K. M. MASUM HABIB AND ROGER K. LAKE

PHYSICAL REVIEW B 86, 045418 (2012) II. METHOD

In this study, four different types of calculations are performed. (i) Geometry optimization and (ii) band structure calculations are performed using DFT. (iii) The electronic transport of the xGNR is calculated using the NEGF formalism coupled with DFT. (iv) The numerical results are explained using analytical expressions for the wave functions in a π orbital basis. The calculation methods and the device structure are discussed below. A. Device structure

FIG. 1. Atomistic geometry of the crossbar GNR (xGNR) consisting of two H-passivated armchair GNRs with one placed on top of the other rotated by 90◦ . Each GNR is 14-C atomic layers wide (∼1.8 nm) with a band gap of 130 meV. The contacts are modeled by the self-energies of semi-infinite leads. The region bounded by the broken lines is used as a supercell for the band structure calculations.

∼1.8 nm × 1.8 nm and a twist angle of 90◦ . For two infinite sheets, a 90◦ rotation is the same as a 30◦ rotation which is not a commensurate rotation angle. A Moiré pattern can be observed at the intersection of the two nanoribbons in Fig. 1. Calculations, based on ab initio density functional theory (DFT) coupled with the nonequilibrium Green’s function formalism (NEGF), show that the interlayer decoupling still exists in such a small geometry containing approximately 220 C atoms leading to strong suppression of inter-GNR transmission when the two layers are at the same potential. An analytical model using Fermi’s golden rule reveals that the suppression of the interlayer transmission results from the cancellation of the quantum phases of the electronic wave functions of the individual GNRs. An external bias applied between the GNRs changes the relative phases of the wave functions resulting in modulation of the transmission and current by several orders of magnitude. The decoupling that occurs when the GNRs are at equal potentials can be exploited using a built-in potential similar to the one that occurs in a p-n junction to produce negative differential resistance with a large (>1000) peak-to-valley current ratio. A large, dense array of crossed graphene nanoribbons, with each cross point providing a nonlinear current-voltage response, could serve in, for example, a cellular neural network,33 a memory array,34 or provide added functionality to standard transistor circuits.35 While in this paper, we consider a two-terminal configuration, one could also control the interlayer potential with gates, in which case the physics described here could be exploited to implement ultrascaled transistors.

The crossbar structure consists of two H-passivated, armchair GNRs shown in Fig. 1. In this arrangement, the GNR along the y axis is placed on top of the GNR along the x axis ˚ in between. Throughout with a vertical separation of 3.35 A the rest of the paper, the GNRs placed along x and y axes will be referred to as the “bottom” and “top” GNRs, respectively. Since we are interested in current modulation in the absence of a band gap, the widths of the GNRs are chosen to be 14-C atomic layers (3n + 2) ∼ 1.8 nm to minimize the band gap resulting from the finite width. The band gap of the 14-aGNR, calculated from DFT code FIREBALL36,37 is 130 meV, which is in good agreement with Son et al.38 The area of the overlap region of the xGNR is ∼1.8 nm × 1.8 nm. The total simulated area between the four ideal leads indicated by the self-energies in Fig. 1 is ∼7 nm × 7 nm. The infinite xGNR, shown in Fig. 1, is constructed by t t attaching the self-energies and to the top GNR and the self-energies b and b to the bottom GNR. Throughout this article, the semi-infinite leads indicated by the self-energies t and b are termed as top and bottom contacts respectively. B.

FIREBALL

The geometry optimization and the calculation of electronic structures are performed with the ab initio quantum mechanical molecular dynamics, DFT code FIREBALL36,39 using separable, nonlocal Troullier-Martins pseudopotentials,40 the BLYP exchange correlation functional41,42 and a self-consistent generalization of the Harris-Foulkes energy functional.43–47 A single zeta (single numeric) sp3 FIREBALL basis set is used. These localized pseudoatomic orbitals are slightly excited due to hard wall boundary conditions imposed at radial cutoffs, ˚ for rc , for each atomic species. The cutoffs are r 1s = 4.10 A ˚ and r 2p = 4.8 A ˚ for carbon.48 hydrogen and r 2s = 4.4 A, C. Structure relaxation

In order to construct the crossbar, the geometry of a supercell of H-passivated single layer aGNR with periodic boundary conditions is optimized using FIREBALL. The supercell, which has a length of eight atomic layers, is repeated ˚ The relaxation is using the lattice vector a = 8.77xˆ (A). performed until all the Cartesian forces on the atoms are ˚ −1 . In the self-consistent field calculation, a Fermi 0.19 eV, m = 10 and n = 10. The AB and BA components are very small at low energies near E = 0 compared to the AA and BB components in both cases. The vertical lines represent the chemical potentials of the contacts.

and its enhancement by the applied bias. Hence the matrix element governs the voltage dependence of the transmission, and we shall concentrate only on Mmn below. In the discussion below, we shall only consider the fundamental modes and hence drop the subscript of M. The matrix element consists of four components, M = M AA + M BB + M AB + M BA as given by Eq. (12). These four components plotted in Fig. 8 are labeled as “AA,” “BB,” “AB,” and “BA.” At low energies, M ≈ M AA + M BB , since M AA and M BB are orders of magnitude larger than M AB and M BA for all bias voltages. The cancellation of the phases of M AA and M BB suppresses the matrix element and hence the transmission at V = 0 V. This can be understood by looking at the phasor diagram, Fig. 9(a), where the matrix element and its four components are shown in polar coordinates at the conduction band edge. The magnitude of M is very small since |M AA | ≈ |M BB | and

M AA − M BB ≈ 180◦ . The applied bias does not change |M AA | and |M BB |, but it modulates the phase difference between these components which, in turn, results in a significant change in the magnitude of the total matrix element, M. For example, at V = 0.25 V, the magnitudes of M AA and M BB shown in Fig. 8(b) remain unchanged. Although |M AB | and |M BA | increase by an order of magnitude, they are still several orders of magnitude smaller compared to |M AA | and |M BB | and hence insignificant. The applied bias changes M BB by ∼60◦ , while leaving M AA almost unchanged as shown in Fig. 9(b). Thus a bias voltage of 0.25 V changes the phase difference, M AA − M BB from 180◦ to ∼120◦ . As a consequence, the total matrix element M ≈ M AA + M BB and the resulting transmission increase by several orders of magnitude.

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αβ αβ αβ FIG. 9. (Color online) Phasor plots of Mmn (a,b), Hmn (c,d), and Cmn (e,f). The AA, BB, AB, and BA components are indicated according to the legend at the top of the figure. (a), (c), and (e) V = 0 V and E = 0.064 eV (the conduction band edge). (b), (d), and (f) V = 0.25 V and E = 0 eV. The lengths and the directions of the arrows represent the magnitude and the angle of the corresponding complex quantities, respectively. Since the AB and BA components of H and M are very small, they are magnified several orders of magnitude and shown in the insets.

The voltage modulation of the phases of the major components M αα = 12 C αα H αα is controlled by the voltage dependent quantum phase factors C αα defined by Eqs. (14) and (17). Figures 9(c) and 9(d) clearly show that the phases of H BB and H AA are only slightly modified by the bias. On the other hand, it is clear from Figs. 9(e) and 9(f) that the bias changes the phase of C BB by ∼60◦ . The phase of C AA remains unchanged for all energies and for all biases due to the particular construction of the wave function given by Eq. (8). Thus the voltage dependency of the quantum phases are lumped into the quantity C BB . The asymmetry in transmission at zero bias results from the phase factors C AB and C BA and the small difference between |M AA | and |M BB |. At the conduction band edge, n = 10 and hence C AB = sgn(n)e−i nkx = − nkx , where

nkx is a small angle. At the valance band edge, n = −10 and hence C AB = sgn(n)e−i nkx = 180◦ − nkx . Thus C AB and hence M AB at the conduction and valence band edges differ by 180◦ . The same is true for the phase of C BA and M BA . The sum of M AB and M BA adds to M AA at the conduction band edge as shown in Fig. 9(a), and the sum adds to M BB at the valence band edge. |M AA | is slightly larger than |M BB |. At the valence band edge, the addition of (M AB + M BA ) to M BB gives a better cancellation with M AA resulting in the matrix element minimum shown in Fig. 8(a). At the conduction band edge, the addition of (M AB + M BA ) to M AA reduces the cancellation with M BB resulting in a larger total matrix element and increased transmission.

In a preliminary study of the sensitivity of the transport properties to the detailed geometry of the overlap region, we have considered four variations of the xGNR shown in Fig. 1: the x and the y coordinates of the top GNR are shifted by (a) acc /2 and (b) 3acc /2, (c) the width of both arms are increased to 4.5 nm (38 atomic C layers), and (d) the width of the top GNR is increased to 20 C atoms so that the xGNR consists of a 14-aGNR and a 20-aGNR. Calculations, based on the model presented in Sec. II F, show that the transport properties of all of these xGNRs are similar to that of the crossbar shown in Fig. 1. The I -V characteristics of these xGNRs with the biasing scheme described in Sec. III B2 are all similar to the I -V ’s shown in Fig. 6. The peak-to-valley current ratios for the (a), (b), (c), and (d) configurations at φbi = 0.25 V are 100, 1000, 150, and 120, respectively. V. CONCLUSIONS

We have performed ab initio DFT and NEGF based calculations to study the inter-layer coupling and transport properties of nanometer scale twisted bilayer graphene that occurs in the overlap region of a crossbar consisting of two GNRs with one placed on top of the other at right angles. The GNRs in the crossbar are electronically decoupled from each other similar to the decoupling that occurs in twisted bilayer graphene. An analytical model based on Fermi’s golden rule reveals that the decoupling is a consequence of the cancellation of quantum phases of the electronic states of the individual

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GNRs. This leads to strong suppression of the inter-GNR transmission when the two GNRs are at the same potential. A potential difference between the GNRs changes the relative phases of the top and bottom wave functions and destroys the phase cancellation resulting in strong coupling and high transmission. Thus the transmission can be modulated several orders of magnitude by controlling the quantum phase using an external bias. A built-in potential between the two GNRs can lead to large peak-to-valley current ratios (>1000) resulting from the strong electronic decoupling of the two GNRs that occurs when they are driven to the same potential. Current

switching by voltage control of the quantum phase in graphene crossbar structure is a novel switching mechanism. It is robust even with an overlap of ∼1.8 nm × 1.8 nm containing only ∼220 C atoms that is well below the smallest horizontal length scale envisioned in the ITRS.

*

23

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ACKNOWLEDGMENT

This work is supported by the Microelectronics Advanced Research Corporation Focus Center on Nano Materials (FENA).

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